DETERMINANTS OF VECTORS AND MATRICES Eric Lehman revised 5th May 2005 1 DETERMINANTS OF VECTORS AND MATRICES Contents 1 Some words about linear algebra 4 1.1 Linear functions (or maps) ............................ 4 1.2 Linear spaces ................................... 4 1.3 Examples of linear maps ............................. 5 1.4 Dimension and basis ............................... 7 2 Geometrical meanings of determinants 9 2.1 Determinant of a square matrix .......................... 9 2.2 Area of a parallelogram .............................. 9 ~ ~ ~ ~ 2.3 Computation of det(~i;~j)(~u;~v) when ~u = ai + cj and ~v = bi + dj ........ 10 2.4 Determinant of a linear function from E to E .................. 12 2.5 Two interpretations of the 2 £ 2 matrix determinant ............... 12 2.6 Back to the linear system ............................. 13 2.7 Order 3 ...................................... 14 2.8 Oriented volume of a parallelepiped constructed on 3 vectors .......... 14 2.9 Solving a system of 3 linear equations in 3 unknowns .............. 16 3 How to compute determinants of matrices 17 3.1 One inductive rule ................................. 17 3.2 Main properties of determinants ......................... 18 3.3 Examples ..................................... 19 4 Definitions and proofs 22 4.1 Determinant of vectors with respect to a basis .................. 22 4.2 Determinant of a linear map from E to E .................... 24 4.3 Determinant of a square matrix .......................... 25 4.4 A beautiful formula ................................ 27 4.5 Linear systems .................................. 27 Preface Determinants are important tools in analysing and solving systems of linear equations. We have for instance: Theorem. A system of n linear equations in n unknowns (x1; x2; : : : ; xn) 8 > a11x1 + a12x2 + ¢ ¢ ¢ + a1nxn = b1 <> a21x1 + a22x2 + ¢ ¢ ¢ + a2nxn = b2 . > . :> an1x1 + an2x2 + ¢ ¢ ¢ + annxn = bn has a unique solution if and only if the determinant of the coefficient matrix 0 1 a11 a12 ¢ ¢ ¢ a1n B C Ba21 a22 ¢ ¢ ¢ a2n C A = (aij)n£n =B . C @ . .. A an1 an2 ¢ ¢ ¢ ann denoted by det A, det(A) or jAj, is different from zero: det A 6= 0: 1 Some words about linear algebra 1.1 Linear functions (or maps) A map f : E ! F , from a set E to a set F is said to be linear if for any ~u and ~v in E: f(~u + ~v) = f(~u) + f(~v) But then we must also have f(~u + ~u) = f(~u) + f(~u) or f(2~u) = 2f(~u) and in the same way f(3~u) = f(~u + 2~u) = f(~u) + f(2~u) = f(~u) + 2f(~u) = 3f(~u); and so on. 1 1 Also, since 2~u + 2~u = ~u, we have 1 1 1 1 f( 2~u) + f( 2~u) = f( 2~u + 2~u) = f(~u) 1 1 1 or 2f( 2~u) = f(~u) or even better f( 2~u) = 2 f(~u). p p p In that way we get for any rational number q the relation f( q ~u) = q f(~u). Since every real number ¸ is a limit of rational numbers, we get the general rule f(¸~u) = ¸f(~u): Now we have a better definition of a linear map or function: A function f : E ! F is linear if for any ~u and ~v in E and any real ¸: ½ f(~u + ~v) = f(~u) + f(~v) f(¸~u) = ¸f(~u) 1.2 Linear spaces Our previous definition of a linear map or function is not yet quite good since we cannot be sure that the addition and the multiplication by a real number have any meaning in the sets E and F . We have to assume that these operations have meaning in the sets E and F , that is to say we must have a specific structure on these sets. The structure we need is called ”linear space”. More precisely: Definition 1.2.1 A set E is called a linear space or a vector space (the elements of E will be called elements or vectors), if there are two operations defined on E: the addition + such that: 8 ~u;~v and ~w: (~u + ~v) + ~w = ~u + (~v + ~w) 9 ~0 such that 8 ~u: ~u + ~0 = ~u 8 ~u; 9 an opposite ¡~u such that ~u + (¡~u) = ~0 8 ~u and ~v: ~u + ~v = ~v + ~u the multiplication of a vector by a real number is such that for any vectors ~u and ~v and for any real numbers ¸ and ¹: 1 SOME WORDS ABOUT LINEAR ALGEBRA 5 1~u = ~u and 0~u = ~0 and (¡1)~u = ¡~u ¸(~u + ~v) = ¸~u + ¸~v (¸ + ¹)~u = ¸~u + ¹~u ¸(¹~u) = (¸¹)~u Example 1.2.2 n = 0 A set with only the null vector ~0 is a vector space if we put the rules ~0 + ~0 = ~0 and ¸~0 = ~0: Example 1.2.3 n = 1 The set of real numbers is a vector space with the usual addition and multiplication. Example 1.2.4 n = 2 The set of matrices with 1 column and 2 rows ½µ ¶¯ ¾ a ¯ R2£1 = M (R) = ¯ a; c 2 R 2;1 c ¯ becomes a vector space if we define addition and multiplication by a real number in the follow- ing ”natural” way: for any two vectors in R2£1: µ ¶ µ ¶ µ ¶ a b a + b + := ; c d c + d and for any vector in R2£1 and for any real number ¸: µ ¶ µ ¶ a ¸a ¸ := : c ¸c The nul vector is then µ ¶ 0 ~0 := : 0 n Example 1.2.5 n The set R or Mn;1(R) is a vector space. m£n Example 1.2.6 For any positive integers m and n, the set of matrices R = Mn;m(R) with n rows and m columns is a vector space. Example 1.2.7 The set of all real functions defined on R is a vector space if f + g is the function x 7! f(x) + g(x) and ¸f is the function x 7! ¸f(x). 1.3 Examples of linear maps a) f : R ! R Proposition 1.3.1 A function f from R to R is linear if and only if there is a real number a such that for any real number u we have: f(u) = au 1 SOME WORDS ABOUT LINEAR ALGEBRA 6 Proof. If there is such an a, we have f(u + v) = a(u + v) = au + av = f(u) + f(v); f(¸u) = a(¸u) = (a¸)u = (¸a)u = ¸(au) = ¸f(u): Conversely, suppose f is linear. Since u = 1u = u1, we get f(u) = f(u1) = uf(1) = f(1)u. Let us put a := f(1), we have for any u in R: f(u) = au. 2 b) f : R2£1 ! R2£1 Proposition 1.3.2 A function f from R2£1 to R2£1 is linear if and only if there is a 2 by 2 matrix A such that for any vector ~u we have: f(~u) = A~u: Proof. First suppose there is such a matrix A. Let us write explicitly A and two vectors ~u and ~v as µ ¶ µ ¶ µ ¶ a b u v A = ; ~u = 1 ; ~v = 1 : c d u2 v2 Then we have µ ¶ µ ¶µ ¶ µ ¶ u a b u au + bu f(~u) = f 1 = 1 = 1 2 : u2 c d u2 cu1 + du2 We can check that f(~u + ~v) = f(~u) + f(~v) explicitly by computing both sides of that equality: µµ ¶ µ ¶¶ µ ¶ µ ¶ µ ¶ u v u +v a(u +v )+b(u +v ) au +bu +av +bv f(~u+~v) = f 1 + 1 = f 1 1 = 1 1 2 2 = 1 2 1 2 u2 v2 u2+v2 c(u1+v1)+d(u2+v2) cu1+du2+cv1+dv2 and on the other hand: µ ¶ µ ¶ µ ¶ µ ¶ µ ¶ u v au +bu av +bv au +bu +av +bv f(~u)+f(~v) = f 1 +f 1 = 1 2 + 1 2 = 1 2 1 2 u2 v2 cu1+du2 cv1+dv2 cu1+du2+cv1+dv2 We have also to check that for any real ¸ and any vector ~u, we have f(¸~u) = ¸f(~u): the left side of the equality is in fact µ µ ¶¶ µ ¶ µ ¶ µ ¶ u ¸u a¸u +b¸u ¸(au +bu ) f(¸~u) = f ¸ 1 = f 1 = 1 2 = 1 2 ; u2 ¸u2 c¸u1+d¸u2 ¸(cu1+du2) and the right hand side is: µ ¶ µ ¶ µ ¶ u au +bu ¸(au +bu ) ¸f(~u) = ¸f 1 = ¸ 1 2 = 1 2 ; u2 cu1+du2 ¸(cu1+du2) Conversely, if f is linear, let us define µ ¶ µ ¶ µ ¶ µ ¶ a 1 b 0 := f ; := f : c 0 d 1 Then we can write for any vector ~u: µ ¶ µ ¶ µ ¶ u1 1 0 ~u = = u1 + u2 ; u2 0 1 1 SOME WORDS ABOUT LINEAR ALGEBRA 7 and thus by linearity µ µ ¶ µ ¶¶ µ µ ¶¶ µ µ ¶¶ µ ¶ µ ¶ 1 0 1 0 1 0 f(~u) = f u + u = f u + f u = u f + u f 1 0 2 1 1 0 2 1 1 0 2 1 µ ¶ µ ¶ µ ¶ µ ¶ µ ¶ µ ¶ µ ¶µ ¶ a b u1a u2b u1a+u2b au1+bu2 a b u1 = u1 + u2 = + = = = c d u1c u2d u1c+u2d cu1+du2 c d u2 2 c) f : R2£1 ! R Proposition 1.3.3 A function f from R2£1 to R is linear if and only if there is a 1 by 2 matrix (a b) such that for any vector ~u we have: ¡ ¢ f(~u) = a b ~u Proof. Left as an exercise. 2 1.4 Dimension and basis a) Affine plane versus vector plane A usual plane such as the blackboard or a sheet of paper gives the image of an affine plane in which all the elements, called points, play the same role.